6.8.10 4.4

6.8.10.1 [1821] Problem 1

problem number 1821

Added Oct 10, 2019.

Problem Chapter 8.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{w}_y+b{w}_z=c{\coth }^n(\beta x)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*D[w[x, y,z], y] +  b*D[w[x,y,z],z]== c*Coth[beta*x]^n*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to \exp \left(\frac{c{\coth }^{n+1}(\beta x){}_2{F}_1(1,\frac{n+1}{2};\frac{n+3}{2};{\coth }^2(\beta x))}{\beta n+\beta }\right)c_1(y-ax,z-bx)\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)= c*coth(beta*x)^n*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(-ax+y,-xb+z){\mathrm{e}}^{\int c{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta x\right)\right)}^n\mathrm{d}x}$$

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6.8.10.2 [1822] Problem 2

problem number 1822

Added Oct 10, 2019.

Problem Chapter 8.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b{w}_y+c\coth (\lambda x)w_z=(k\coth (\beta x)+s\coth (\gamma z))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +  c*Coth[lambda*x]*D[w[x,y,z],z]== (k*Coth[beta*x]+s*Coth[gamma*z])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-\frac{bx}{a},z-\frac{c\log (\sinh (\lambda x))}{a\lambda })\exp ({\int }_1^x\frac{k\coth (\beta K[1])+s\coth \left(\frac{\gamma (a\lambda z-c\log (\sinh (\lambda x))+c\log (\sinh (\lambda K[1])))}{a\lambda }\right)}{a}dK[1])\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x, y,z), x) + b*diff(w(x, y,z), y) +  c*coth(lambda*x)*diff(w(x,y,z),z)= (k*coth(beta*x)+s*coth(gamma*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(\frac{ay-xb}{a},\frac{2za\lambda +c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)-1)+c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)+1)}{2a\lambda }){\mathrm{e}}^{-{\int }^x\frac{1}{a}(-k\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta \mathit{\_}\mathit{a}\right)+s\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\frac{\gamma (-2za\lambda -c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)-1)-c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)+1)+\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathit{\_}\mathit{a}\lambda \right)-1)c+\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathit{\_}\mathit{a}\lambda \right)+1)c)}{2a\lambda }\right))d\mathit{\_}\mathit{a}}$$

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6.8.10.3 [1823] Problem 3

problem number 1823

Added Oct 10, 2019.

Problem Chapter 8.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{\coth }^n(\beta x)w_y+b{\coth }^k(\lambda x)w_z=c{\coth }^m(\gamma x)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Coth[beta*x]^n*D[w[x, y,z], y] +  b*Coth[lambda*x]^k*D[w[x,y,z],z]== c*Coth[gamma*x]^m *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to \exp \left(\frac{c{\coth }^{m+1}(\gamma x){}_2{F}_1(1,\frac{m+1}{2};\frac{m+3}{2};{\coth }^2(\gamma x))}{\gamma m+\gamma }\right)c_1(z-\frac{b{\coth }^{k+1}(\lambda x){}_2{F}_1(1,\frac{k+1}{2};\frac{k+3}{2};{\coth }^2(\lambda x))}{k\lambda +\lambda },y-\frac{a{\coth }^{n+1}(\beta x){}_2{F}_1(1,\frac{n+1}{2};\frac{n+3}{2};{\coth }^2(\beta x))}{\beta n+\beta })\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x, y,z), x) + a*coth(beta*x)^n*diff(w(x, y,z), y) +  b*coth(lambda*x)^k*diff(w(x,y,z),z)= c*coth(gamma*x)^m *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(-\int a{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta x\right)\right)}^n\mathrm{d}x+y,-\int b{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)\right)}^k\mathrm{d}x+z){\mathrm{e}}^{\int c{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\gamma \right)\right)}^m\mathrm{d}x}$$

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6.8.10.4 [1824] Problem 4

problem number 1824

Added Oct 10, 2019.

Problem Chapter 8.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b\coth (\beta y)w_y+c\coth (\lambda x)w_z=k\coth (\gamma z)w$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Coth[beta*y]*D[w[x, y,z], y] +  c*Coth[lambda*x]*D[w[x,y,z],z]== k*Coth[gamma*z] *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x, y,z), x) + b*coth(beta*y)*diff(w(x, y,z), y) +  c*coth(lambda*x)*diff(w(x,y,z),z)= k*coth(gamma*z) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(\frac{1}{2\beta b}(-2b\beta x+a\ln \left(\frac{{(\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-1)}^2}{\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-2}\right)-\ln \left(\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))\right)a),\frac{2za\lambda +c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)-1)+c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)+1)}{2a\lambda }){\mathrm{e}}^{-{\int }^x\frac{k}{a}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\frac{\gamma (-2za\lambda -c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)-1)-c\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)+1)+\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathit{\_}\mathit{a}\lambda \right)-1)c+\ln (\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathit{\_}\mathit{a}\lambda \right)+1)c)}{2a\lambda }\right)d\mathit{\_}\mathit{a}}$$

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6.8.10.5 [1825] Problem 5

problem number 1825

Added Oct 10, 2019.

Problem Chapter 8.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b\coth (\beta y)w_y+c\coth (\gamma z)w_z=k\coth (\lambda x)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Coth[beta*y]*D[w[x, y,z], y] +  c*Coth[gamma*z]*D[w[x,y,z],z]== k*Coth[lambda*x] *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{k(\log (-\tanh (\lambda x))+\log (\cosh (\lambda x)))}{a\lambda }}{c}_1(\frac{a\log (\text{sech}(\beta y))+b\beta x}{2a\beta },\frac{2c\log (\text{sech}(\beta y))}{\beta }-\frac{b\log ({\text{sech}}^2(\gamma z))}{\gamma })\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x, y,z), x) + b*coth(beta*y)*diff(w(x, y,z), y) +  c*coth(gamma*z)*diff(w(x,y,z),z)= k*coth(lambda*x) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(\frac{1}{2\beta b}(-2b\beta x+a\ln \left(\frac{{(\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-1)}^2}{\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-2}\right)-\ln \left(\mathrm{RootOf}(\beta y-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))\right)a),\frac{1}{2c\gamma }(-2c\gamma x+a\ln \left(\frac{{(\mathrm{RootOf}(\gamma z-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-1)}^2}{\mathrm{RootOf}(\gamma z-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))-2}\right)-\ln \left(\mathrm{RootOf}(\gamma z-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(\mathit{\_}\mathit{Z}-1))\right)a)){(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)-1)}^{-\frac{k}{2a\lambda }}{(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(x\lambda \right)+1)}^{-\frac{k}{2a\lambda }}$$

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6.8.10.6 [1826] Problem 6

problem number 1826

Added Oct 10, 2019.

Problem Chapter 8.4.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a_1{\coth }^{n_1}({\lambda }_{1}x)w_x+b_1{\coth }^{m_1}({\beta }_{1}y)w_y+c_1{\coth }^{k_1}({\gamma }_{1}z)w_z=(a_2{\coth }^{n_2}({\lambda }_{2}x)w_x+b_2{\coth }^{m_2}({\beta }_{2}y)w_y+c_2{\coth }^{k_2}({\gamma }_{2}z))w$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Coth[lambda1*x]^n1*D[w[x, y,z], x] + b1*Coth[beta1*y]^m1*D[w[x, y,z], y] +  c1*Coth[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Coth[lambda2*x]^n2+b2*Coth[beta2*y]^m2+c2*Coth[gamma2*x]^k2) *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a1*coth(lambda1*x)^n1*diff(w(x, y,z), x) + b1*coth(beta1*y)^m1*diff(w(x, y,z), y) +  c1*coth(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*coth(lambda2*x)^n2+b2*coth(beta2*y)^m2+c2*coth(gamma2*x)^k2) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(-\int {\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\lambda 1x\right)\right)}^{-\mathit{n}\mathit{1}}\mathrm{d}x+\int \frac{{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta 1y\right)\right)}^{-\mathit{m}\mathit{1}}\mathit{a}\mathit{1}}{\mathit{b}\mathit{1}}\mathrm{d}y,\frac{1}{\mathit{a}\mathit{1}}(z\mathit{a}\mathit{1}-\mathit{c}\mathit{1}\int {\left(\frac{\cosh \left(\lambda 1x\right)}{\sinh \left(\lambda 1x\right)}\right)}^{-\mathit{n}\mathit{1}}{\left(\frac{\cosh \left(\gamma 1x\right)}{\sinh \left(\gamma 1x\right)}\right)}^{\mathit{k}\mathit{1}}\mathrm{d}x)){\mathrm{e}}^{{\int }^x\frac{{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\lambda 1\mathit{\_}\mathit{f}\right)\right)}^{-\mathit{n}\mathit{1}}}{\mathit{a}\mathit{1}}(\mathit{a}\mathit{2}{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\lambda 2\mathit{\_}\mathit{f}\right)\right)}^{\mathit{n}\mathit{2}}+\mathit{c}\mathit{2}{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\gamma 2\mathit{\_}\mathit{f}\right)\right)}^{\mathit{k}\mathit{2}}+\mathit{b}\mathit{2}{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta 2\mathrm{RootOf}(\int {\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\lambda 1\mathit{\_}\mathit{f}\right)\right)}^{-\mathit{n}\mathit{1}}\mathrm{d}\mathit{\_}\mathit{f}-{\int }^{\mathit{\_}\mathit{Z}}\frac{{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta 1\mathit{\_}\mathit{a}\right)\right)}^{-\mathit{m}\mathit{1}}\mathit{a}\mathit{1}}{\mathit{b}\mathit{1}}d\mathit{\_}\mathit{a}-\int {\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\lambda 1x\right)\right)}^{-\mathit{n}\mathit{1}}\mathrm{d}x+\int \frac{{\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\beta 1y\right)\right)}^{-\mathit{m}\mathit{1}}\mathit{a}\mathit{1}}{\mathit{b}\mathit{1}}\mathrm{d}y)\right)\right)}^{\mathit{m}\mathit{2}})d\mathit{\_}\mathit{f}}$$

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